= Ground-state transform for a weighted Dirichlet energy
Let $\Phi\in C^2(I)$ on an <open interval> $I$, take a real smooth <function> $h$ of <compact support> in $I$, and put $w=e^{-\Phi}$ and $g=he^{-\Phi/2}$. Expanding $h'e^{-\Phi/2}=g'+\tfrac12\Phi'g$ and using <integration by parts> gives
$$
\int_I|h'|^2e^{-\Phi}=\int_I|g'|^2+\int_I\left(\frac{|\Phi'|^2}{4}-\frac{\Phi''}{2}\right)|g|^2.
$$
This transforms the weighted gradient energy into the quadratic form of a <Schrödinger operator> with potential $|\Phi'|^2/4-\Phi''/2$.
On a finite interval $[a,b]$, without boundary vanishing the right-hand side additionally contains
$$
\frac12\bigl[\Phi'|g|^2\bigr]_a^b.
$$
Thus the boundary-free identity requires compact support, suitable zero <boundary conditions>, or sufficient endpoint decay. The corresponding infinite-interval statement also requires convergence of the integrals and disappearance of the boundary term.
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